cp's OEIS Frontend

A137725 Number of sequences of length n with elements {-2,-1,+1,+2}, such that the sum of elements of the whole sequence but of no proper subsequence equals 0 modulo n. For n>=4, the number of Hamiltonian (directed) circuits on the circulant graph C_n(1,2).

%I A137725 #25 Jul 13 2025 11:01:58
%S A137725 4,4,16,18,24,32,46,58,82,112,158,220,316,450,650,938,1364,1982,2892,
%T A137725 4220,6170,9022,13206,19332,28314,41472,60760,89022,130446,191150,
%U A137725 280120,410506,601600,881656,1292102,1893634,2775226,4067256,5960822,8735972,12803156,18763898,27499794,40302866,59066684
%N A137725 Number of sequences of length n with elements {-2,-1,+1,+2}, such that the sum of elements of the whole sequence but of no proper subsequence equals 0 modulo n. For n>=4, the number of Hamiltonian (directed) circuits on the circulant graph C_n(1,2).
%C A137725 Number of 1-D walks with jumps to next-nearest neighbors with n steps, starting at 0 and ending at -2n, -n, 0, n, or 2n, such that every point is visited at most once and every pair of points at the distance n contains at least one unvisited point (not counting the ending visit). Cf. A092765.
%C A137725 For n>1, the number of circular permutations (counted up to rotations) of {0, 1,...,n-1} such that the distance between every two adjacent elements is -2,-1,1,or 2 modulo n. Cf. A003274.
%H A137725 G. C. Greubel, <a href="/A137725/b137725.txt">Table of n, a(n) for n = 1..1000</a>
%H A137725 Mordecai J. Golin and Yiu Cho Leung, <a href="http://www.cse.ust.hk/tcsc/RR/2004-02.ps.gz">Unhooking Circulant Graphs: A Combinatorial Method for Counting Spanning Trees, Hamiltonian Cycles and other Parameters</a>, Technical report HKUST-TCSC-2004-02. See <a href="https://cse.hkust.edu.hk/~golin/pubs/golin_WG04.pdf">also</a>
%H A137725 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CirculantGraph.html">Circulant Graph</a>
%H A137725 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HamiltonianCycle.html">Hamiltonian Cycle</a>
%H A137725 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1,0,-1,1).
%F A137725 For even n>=4, a(n) = 2*(n + 3*A000930(n) - 2*A000930(n-1)); for odd n>=3, a(n) = 2*(n + 1 + 3*A000930(n) - 2*A000930(n-1)).
%F A137725 For n>8, a(n) = 2*a(n-1) - a(n-3) - a(n-5) + a(n-6) or a(n) = a(n-1) + a(n-2) - a(n-5) - 4.
%F A137725 O.g.f.: -2*x^2-2*x-6-1/(x+1)+2/(x-1)^2+1/(x-1)+(4*x-6)/(x^3+x-1). - _R. J. Mathar_, Feb 10 2008
%t A137725 CoefficientList[Series[-2*x^2-2*x-6-1/(x+1)+2/(x-1)^2+1/(x-1)+(4*x-6)/(x^3+x-1), {x, 0, 50}], x] (* _G. C. Greubel_, Apr 28 2017 *)
%o A137725 (PARI) my(x='x+O('x^50)); Vec(-2*x^2-2*x-6-1/(x+1)+2/(x-1)^2+1/(x-1)+(4*x-6)/(x^3+x-1)) \\ _G. C. Greubel_, Apr 28 2017
%Y A137725 Cf. A124353, A137726.
%K A137725 nonn
%O A137725 1,1
%A A137725 _Max Alekseyev_, Feb 08 2008
%E A137725 Typo in formulas corrected by _Max Alekseyev_, Nov 03 2010